Chapter 5 Quadrilaterals

1
Chapter 5Quadrilaterals

• Apply the definition of a parallelogram
• Prove that certain quadrilaterals are
  parallelograms
• Apply the theorems and definitions about the
  special quadrilaterals

2
5-1 Properties of Parallelograms

• Objectives
• Apply the definition of a parallelogram
• List the other properties of a parallelogram
  through new theorems

3
Quadrilaterals

• Any 4 sided figure

4
Definition of a Parallelogram ( )

• If the opposite sides of a quadrilateral are
  parallel, then it is a parallelogram.

ABCD
5
Naming a Parallelogram

• Use the symbol for parallelogram and name
  using the 4 vertices in order either clockwise or
  counter clockwise.

ABCD
6
Theorem

• Opposite sides of a parallelogram are congruent.

A
B
D
C
7
Theorem

• Opposite angles of a parallelogram are congruent.

A
B
D
C
8
Theorem

• The diagonals of a parallelogram bisect each
  other.

A
B
D
C
9
Remote Time

• True or False

10
True or False

• Every parallelogram is a quadrilateral

11
True or False

• Every quadrilateral is a parallelogram

12
True or False

• All angles of a parallelogram are congruent

13
True or False

• All sides of a parallelogram are congruent

14
True or False

• In RSTU, RS TU.
• ? Hint draw a picture

15
True or False

• In ABCD, if m ? A 50, then m ? C 130.
• ? Hint draw a picture

16
True or False

• In XWYZ, XY ?WZ
• ? Hint draw a picture

17
True or False

• In ABCD, AC and BD bisect each other
• ? Hint draw a picture

18
White Board Practice

• Given ABCD
• Name all pairs of parallel sides

19
White Board Practice

• Given ABCD
• AB DC
• BC AD

20
White Board Practice

• Given ABCD
• Name all pairs of congruent angles

21
White Board Practice

• Given ABCD
• ? BAD ? ? DCB ? CBD ? ? ADB
• ? ABC ? ? CDA ? ABD ? ? CDB
• ? BEA ? ? DEC ? BCA ? ? DAC
• ? BEC ? ? DEA ? BAC ? ? DCA

22
White Board Practice

• Given ABCD
• Name all pairs of congruent segments

23
White Board Practice

• Given ABCD
• AB ? CD
• BC ? DA
• BE ? ED
• AE ? EC

24
White Board Groups

• Quadrilateral RSTU is a parallelogram. Find the
  values of x, y, a, and b.

6
R
S
yº
xº
9
b
80º
T
U
a
25
White Board Groups

• Quadrilateral RSTU is a parallelogram. Find the
  values of x, y, a, and b.
• x 80
• y 45
• a 6
• b 9

26
White Board Groups

• Quadrilateral RSTU is a parallelogram. Find the
  values of x, y, a, and b.

R
S
yº
xº
9
12
a
b
45º
35º
U
T
27
White Board Groups

• Quadrilateral RSTU is a parallelogram. Find the
  values of x, y, a, and b.
• x 100
• y 45
• a 12
• b 9

28
White Board Groups

• Given this parallelogram with the diagonals
  drawn.

22
18
4y - 2
2x 8
29
White Board Groups

• Given this parallelogram with the diagonals
  drawn.
• x 5
• y 6

30
5-2Ways to Prove that Quadrilaterals are
Parallelograms

• Objectives
• Learn about ways to prove a quadrilateral is a
  parallelogram

31
Use the Definition of a Parallelogram

• Show that both pairs of opposite sides of a
  quadrilateral are parallel
• Then the quadrilateral is a parallelogram

32
Theorem

• Show that both pairs of opposite sides are
  congruent.
• If both pairs of opposite sides of a
  quadrilateral are congruent, then it is a
  parallelogram.

A
B
D
C
33
Theorem

• Show that one pair of opposite sides are both
  congruent and parallel.
• If one pair of opposite sides of a quadrilateral
  are both congruent and parallel, then it is a
  parallelogram.

A
B
D
C
34
Theorem

• Show that both pairs of opposite angles are
  congruent.
• If both pairs of opposite angles of a
  quadrilateral are congruent, then it is a
  parallelogram.

A
B
D
C
35
Theorem

• Show that the diagonals bisect each other.
• If the diagonals of a quadrilateral bisect each
  other, then it is a parallelogram.

A
B
X
D
C
36
Five ways to prove a Quadrilateral is a
Parallelogram

• Show that both pairs of opposite sides parallel
• Show that both pairs of opposite sides congruent
• Show that one pair of opposite sides are both
  congruent and parallel
• Show that both pairs of opposite angles congruent
• Show that diagonals that bisect each other

37
The diagonals of a quadrilateral _____________
bisect each other

• A. Sometimes
• Always
• Never
• I dont know

38
If the measure of two angles of a quadrilateral
are equal, then the quadrilateral is ____________
a parallelogram

1. Sometimes
2. Always
3. Never
4. I dont know

39
If one pair of opposite sides of a quadrilateral
is congruent and parallel, then the quadrilateral
is ___________ a parallelogram

• A. Sometimes
• B. Always
• C. Never
• D. I dont know

40
If both pairs of opposite sides of a
quadrilateral are congruent, then the
quadrilateral is __________ a parallelogram

• A.) Sometimes
• B.) Always
• C.) Never
• D.) I dont know

41
To prove a quadrilateral is a parallelogram, it
is ________ enough to show that one pair of
opposite sides is parallel.

• A.) Sometimes
• B.) Always
• C.) Never
• D.) I dont know

42
5-3 Theorems Involving Parallel Lines

• Objectives
• Apply the theorems about parallel lines and
  triangles

43
Theorem

• If two lines are parallel, then all points on one
  line are equidistant from the other.

m
n
44
Theorem

• If three parallel lines cut off congruent
  segments on one transversal, then they do so on
  any transversal.

A
D
B
E
C
F
45
Theorem

• A line that contains the midpoint of one side of
  a triangle and is parallel to a another side
  passes through the midpoint of the third side.

A
X
Y
B
C
46
Theorem

• A segment that joins the midpoints of two sides
  of a triangle is parallel to the third side and
  its length is half the length of the third side.

A
X
Y
B
C
47
White Board Practice

• Given R, S, and T are midpoint of the sides of
  ? ABC

AB BC AC ST RT RS
12 14 18
15 22 10
10 9 7.8
B
R
S
C
A
T
48
White Board Practice

• Given R, S, and T are midpoint of the sides of
  ? ABC

AB BC AC ST RT RS
12 14 18 6 7 9
20 15 22 10 7.5 11
10 18 15.6 5 9 7.8
B
R
S
C
A
T
49
White Board Practice

• Given that AR BS CT
• RS ? ST

R
S
A
T
B
C
50
White Board Practice

• Given that AR BS CT
• RS ? ST
• If RS 12, then ST ____

R
S
A
T
B
C
51
White Board Practice

• Given that AR BS CT
• RS ? ST
• If RS 12, then ST 12

R
S
A
T
B
C
52
White Board Practice

• Given that AR BS CT
• RS ? ST
• If AB 8, then BC ___

R
S
A
T
B
C
53
White Board Practice

• Given that AR BS CT
• RS ? ST
• If AB 8, then BC 8

R
S
A
T
B
C
54
White Board Practice

• Given that AR BS CT
• RS ? ST
• If AC 20, then AB ___

R
S
A
T
B
C
55
White Board Practice

• Given that AR BS CT
• RS ? ST
• If AC 20, then AB 10

R
S
A
T
B
C
56
White Board Practice

• Given that AR BS CT
• RS ? ST
• If AC 10x, then BC ____

R
S
A
T
B
C
57
White Board Practice

• Given that AR BS CT
• RS ? ST
• If AC 10x, then BC 5x

R
S
A
T
B
C
58
5.4 Special Parallelograms

• Objectives
• Apply the definitions and identify the special
  properties of a rectangle, rhombus and square.

59
Rectangle

• By definition, it is a quadrilateral with four
  right angles.

R
V
S
T
60
Rhombus

• By definition, it is a quadrilateral with four
  congruent sides.

B
C
A
D
61
Square

• By definition, it is a quadrilateral with four
  right angles and four congruent sides.

C
B
D
A
62
Theorem

• The diagonals of a rectangle are congruent.
• WY ? XZ

W
Z
P
X
Y
63
Theorem

• The diagonals of a rhombus are perpendicular.

K
X
J
L
M
64
Theorem

• Each diagonal of a rhombus bisects the opposite
  angles.

K
X
J
L
M
65
Theorem

• The midpoint of the hypotenuse of a right
  triangle is equidistant from the three vertices.

A
X
B
C
66
Theorem

• If an angle of a parallelogram is a right angle,
  then the parallelogram is a rectangle.

R
V
S
T
67
Theorem

• If two consecutive sides of a parallelogram are
  congruent, then the parallelogram is a rhombus.

B
C
A
D
68
White Board Practice

• Quadrilateral ABCD is a rhombus
• Find the measure of each angle
• 1. ? ACD
• 2. ? DEC
• 3. ? EDC
• 4. ? ABC

A
B
E
62º
D
C
69
White Board Practice

• Quadrilateral ABCD is a rhombus
• Find the measure of each angle
• 1. ? ACD 62
• 2. ? DEC 90
• 3. ? EDC 28
• 4. ? ABC 56

A
B
E
62º
D
C
70
White Board Practice

• Quadrilateral MNOP is a rectangle
• Find the measure of each angle
• 1. m ? PON
• 2. m ? PMO
• 3. PL
• 4. MO

M
N
29º
12
L
P
O
71
White Board Practice

• Quadrilateral MNOP is a rectangle
• Find the measure of each angle
• 1. m ? PON 90
• 2. m ? PMO 61
• 3. PL 12
• 4. MO 24

M
N
29º
12
L
P
O
72
White Board Practice

• ? ABC is a right ? M is the midpoint of AB
• 1. If AM 7, then MB ____, AB ____,
• and CM _____ .

A
M
B
C
73
White Board Practice

• ? ABC is a right ? M is the midpoint of AB
• 1. If AM 7, then MB 7, AB 14,
• and CM 7 .

A
M
B
C
74
White Board Practice

• ? ABC is a right ? M is the midpoint of AB
• 1. If AB x, then AM ____, MB _____,
• and MC _____ .

A
M
B
C
75
White Board Practice

• ? ABC is a right ? M is the midpoint of AB
• 1. If AB x, then AM ½ x, MB ½ x,
• and MC ½ x .

A
M
B
C
76
Remote Time

1. Always
2. Sometimes
3. Never
4. I dont know

77
A. AlwaysB. SometimesC. NeverD. I dont
know

• A square is ____________ a rhombus

78
A. AlwaysB. SometimesC. NeverD. I dont
know

• The diagonals of a parallelogram ____________
  bisect the angles of the parallelogram.

79
A. AlwaysB. SometimesC. NeverD. I dont
know

• A quadrilateral with one pairs of sides congruent
  and one pair parallel is _________________ a
  parallelogram.

80
A. AlwaysB. SometimesC. NeverD. I dont
know

• The diagonals of a rhombus are ___________
  congruent.

81
A. AlwaysB. SometimesC. NeverD. I dont
know

• A rectangle ______________ has consecutive sides
  congruent.

82
A. AlwaysB. SometimesC. NeverD. I dont
know

• A rectangle ____________ has perpendicular
  diagonals.

83
A. AlwaysB. SometimesC. NeverD. I dont
know

• The diagonals of a rhombus ___________ bisect
  each other.

84
A. AlwaysB. SometimesC. NeverD. I dont
know

• The diagonals of a parallelogram are ____________
  perpendicular bisectors of eah other.

85
5.5 Trapezoids

• Objectives
• Apply the definitions and learn the properties of
  a trapezoid and an isosceles trapezoid.

86
Trapezoid

• A quadrilateral with exactly one pair of parallel
  sides.

B
C
Trap. ABCD
A
D
87
Anatomy Of a Trapezoid

• The bases are the parallel sides

Base
R
S
T
V
Base
88
Anatomy Of a Trapezoid

• The legs are the non-parallel sides

R
S
Leg
Leg
T
V
89
Isosceles Trapezoid

• A trapezoid with congruent legs.

J
M
K
L
90
Theorem 5-18

• The base angles of an isosceles trapezoid are
  congruent.

F
G
E
H
91
The Median of a Trapezoid

• A segment that joins the midpoints of the legs.

B
C
X
Y
A
D
92
Theorem

• The median of a trapezoid is parallel to the
  bases and its length is the average of the bases.

B
B
C
C
X
X
Y
Y
A
D
A
D
93
White Board Practice

• Complete
• 1. AD 25, BC 13, XY ______

B
C
X
Y
A
D
94
White Board Practice

• Complete
• 1. AD 25, BC 13, XY 19

B
C
X
Y
A
D
95
White Board Practice

• Complete
• 2. AX 11, YD 8, AB _____, DC ____

B
C
X
Y
A
D
96
White Board Practice

• Complete
• 2. AX 11, YD 8, AB 22, DC 16

B
C
X
Y
A
D
97
White Board Practice

• Complete
• 3. AD 29, XY 24, BC ______

B
C
X
Y
A
D
98
White Board Practice

• Complete
• 3. AD 29, XY 24, BC 19

B
C
X
Y
A
D
99
White Board Practice

• Complete
• 4. AD 7y 6, XY 5y -3, BC y 5, y ____

B
C
X
Y
A
D
100
White Board Practice

• Complete
• 4. AD 7y 6, XY 5y -3, BC y 5, y 3.5

B
C
X
Y
A
D
101
Homework Set 5.5

• WS PM 28
• 5-5 1-27 odd
• Quiz next class day